Editing Continuum mechanics (section)

===Balance laws===
Let <math>f(\mathbf{x},t)</math> be a physical quantity that is flowing through the body.  Let <math>g(\mathbf{x},t)</math> be sources on the surface of the body and let <math>h(\mathbf{x},t)</math> be sources inside the body.  Let <math>\mathbf{n}(\mathbf{x},t)</math> be the outward unit normal to the surface <math>\partial \Omega </math>.  Let <math>\mathbf{v}(\mathbf{x},t)</math> be the flow velocity of the physical particles that carry the physical quantity that is flowing.  Also, let the speed at which the bounding surface <math>\partial \Omega </math> is moving be <math>u_n</math> (in the direction <math>\mathbf{n}</math>).

Then, balance laws can be expressed in the general form
:<math>
    \cfrac{d}{dt}\left[\int_{\Omega} f(\mathbf{x},t)~\text{dV}\right] = 
      \int_{\partial \Omega } f(\mathbf{x},t)[u_n(\mathbf{x},t) - \mathbf{v}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x},t)]~\text{dA} + 
      \int_{\partial \Omega } g(\mathbf{x},t)~\text{dA} + \int_{\Omega} h(\mathbf{x},t)~\text{dV} ~.
  </math>
The functions <math>f(\mathbf{x},t)</math>, <math>g(\mathbf{x},t)</math>, and <math>h(\mathbf{x},t)</math> can be scalar valued, vector valued, or tensor valued - depending on the physical quantity that the balance equation deals with.  If there are internal boundaries in the body, jump discontinuities also need to be specified in the balance laws.

If we take the [[Lagrangian and Eulerian specification of the flow field|Eulerian]] point of view, it can be shown that the balance laws of mass, momentum, and energy for a solid can be written as (assuming the source term is zero for the mass and angular momentum equations)
:<math>
    {
    \begin{align}
      \dot{\rho} + \rho (\boldsymbol{\nabla} \cdot \mathbf{v}) & = 0 
          & & \qquad\text{Balance of Mass} \\
      \rho~\dot{\mathbf{v}} - \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} - \rho~\mathbf{b} & = 0 
          & & \qquad\text{Balance of Linear Momentum (Cauchy's first law of motion)} \\
      \boldsymbol{\sigma} & = \boldsymbol{\sigma}^T
          & & \qquad\text{Balance of Angular Momentum (Cauchy's second law of motion)} \\
      \rho~\dot{e} - \boldsymbol{\sigma}:(\boldsymbol{\nabla}\mathbf{v}) + \boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s & = 0
          & & \qquad\text{Balance of Energy.}
    \end{align}
    }
  </math>

In the above equations <math>\rho(\mathbf{x},t)</math> is the mass density (current),   <math>\dot{\rho}</math> is the material time derivative of <math>\rho</math>, <math>\mathbf{v}(\mathbf{x},t)</math> is the particle velocity, <math>\dot{\mathbf{v}}</math> is the material time derivative of <math>\mathbf{v}</math>, <math>\boldsymbol{\sigma}(\mathbf{x},t)</math> is the [[Cauchy stress tensor]], <math>\mathbf{b}(\mathbf{x},t)</math> is the body force density, <math>e(\mathbf{x},t)</math> is the internal energy per unit mass, <math>\dot{e}</math> is the material time derivative of <math>e</math>, <math>\mathbf{q}(\mathbf{x},t)</math> is the heat flux vector, and <math>s(\mathbf{x},t)</math> is an energy source per unit mass. The operators used are defined [[#Operators|below]].
  
With respect to the reference configuration (the Lagrangian point of view), the balance laws can be written as
:<math>
    {
    \begin{align}
      \rho~\det(\boldsymbol{F}) - \rho_0 &= 0 & &  \qquad \text{Balance of Mass} \\
      \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{P} -\rho_0~\mathbf{b} & = 0  & & 
        \qquad \text{Balance of Linear Momentum} \\
      \boldsymbol{F}\cdot\boldsymbol{P}^T & = \boldsymbol{P}\cdot\boldsymbol{F}^T  & & 
        \qquad \text{Balance of Angular Momentum} \\ 
      \rho_0~\dot{e} - \boldsymbol{P}^T:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0
          & & \qquad\text{Balance of Energy.} 
    \end{align}
    }
  </math>

In the above, <math>\boldsymbol{P}</math> is the first [[Piola-Kirchhoff stress tensor]], and   <math>\rho_0</math> is the mass density in the reference configuration.  The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by
:<math>
    \boldsymbol{P} = J~\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}
   ~\text{where}~ J = \det(\boldsymbol{F})
  </math>

We can alternatively define the nominal stress tensor <math>\boldsymbol{N}</math> which is the transpose of the first Piola-Kirchhoff stress tensor such that
:<math>
    \boldsymbol{N} = \boldsymbol{P}^T = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma} ~.
  </math>
Then the balance laws become
:<math>
    {
    \begin{align}
      \rho~\det(\boldsymbol{F}) - \rho_0 &= 0 & &  \qquad \text{Balance of Mass} \\
      \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{N}^T -\rho_0~\mathbf{b} & = 0  & & 
        \qquad \text{Balance of Linear Momentum} \\
      \boldsymbol{F}\cdot\boldsymbol{N} & = \boldsymbol{N}^T\cdot\boldsymbol{F}^T  & & 
        \qquad \text{Balance of Angular Momentum} \\ 
      \rho_0~\dot{e} - \boldsymbol{N}:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0
          & & \qquad\text{Balance of Energy.} 
    \end{align}
    }
  </math>

====Operators====

The operators in the above equations are defined as
:<math>
    \begin{align}
\boldsymbol{\nabla} \mathbf{v}
  &= \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial x_j}\mathbf{e}_i\otimes\mathbf{e}_j
  = v_{i,j}\mathbf{e}_i\otimes\mathbf{e}_j ~; \\[1ex]
\boldsymbol{\nabla} \cdot \mathbf{v}
  &= \sum_{i=1}^3 \frac{\partial v_i}{\partial x_i}
  = v_{i,i} ~; \\[1ex]
\boldsymbol{\nabla} \cdot \boldsymbol{S}
  &= \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial x_j}~\mathbf{e}_i 
  = \sigma_{ij,j}~\mathbf{e}_i ~.
\end{align}
  </math>
where <math>\mathbf{v}</math> is a vector field, <math>\boldsymbol{S}</math> is a second-order tensor field, and <math>\mathbf{e}_i</math> are the components of an orthonormal basis in the current configuration.  Also,
:<math>
    \begin{align}
\boldsymbol{\nabla}_{\circ} \mathbf{v}
  &= \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial X_j}\mathbf{E}_i\otimes\mathbf{E}_j
  = v_{i,j}\mathbf{E}_i\otimes\mathbf{E}_j ~; \\[1ex]
\boldsymbol{\nabla}_{\circ}\cdot\mathbf{v}
  &=  \sum_{i=1}^3 \frac{\partial v_i}{\partial X_i} = v_{i,i} ~; \\[1ex]
\boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{S}
  &= \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial X_j}~\mathbf{E}_i
  = S_{ij,j}~\mathbf{E}_i
\end{align} 
  </math>
where <math>\mathbf{v}</math> is a vector field, <math>\boldsymbol{S}</math> is a second-order tensor field, and <math>\mathbf{E}_i</math> are the components of an orthonormal basis in the reference configuration.

The inner product is defined as
:<math>
    \boldsymbol{A}:\boldsymbol{B} = \sum_{i,j=1}^3 A_{ij}~B_{ij} = \operatorname{trace}(\boldsymbol{A}\boldsymbol{B}^T) ~.
  </math>